I'm in search of the Cartesian equation of Luroth's quartics, i.e. of those curves passing through the 10 vertices of a pentalateral complete.

I have found in Internet a formula of the type

1 / L0*l0 + 1 / L1*l1 + 1 / L2*l2 + 1 / L3*l3 + 1 / L4*l4

but I can't obtain from this formula the Cartesian equation of the quartic equations of Luroth in the form

a * x^4 + b * y^4 + c * x^3 * y + ... + D = 0

Ii think that L0, L1, L2, L3, L4 are of the coefficients of the quartic equations and that l0, l1, l2, l3, l4 are linked to the cartesian equations of the pentalateral, but more than this i don't know ...